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I've heard that General Relativity entails matter (or mass) is necessary for time to exist. However, there are vacuum solutions where the universe is empty of matter but still has spacetime.

P.s: I'm not necessarily talking about proper time. I'm referring to time in general.

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    $\begingroup$ Is not Minkowski space a (much) more elementary counterexample than the de Sitter solution? $\endgroup$ – WillO May 15 at 23:11
  • $\begingroup$ Time and energy are Fourier conjugates (or more generally, spacetime and energy-momentum) and cannot exist in the physical reality without each other. In other words, GR states that spacetime is the field produced by matter just like the electromagnetic field is produced by charges. Vacuum solutions are unphysical, they don’t exist in reality. Their flaw is that the equations are solved without realistic physical initial conditions. This approach and resulting solutions are physically meaningless. $\endgroup$ – safesphere May 16 at 6:47
  • $\begingroup$ @WillO Indeed. And the OP essentially answers his own question. He asks about the existence of time in an empty universe. Well, no such universe exists, whether de Sitter or Minkowski. $\endgroup$ – safesphere May 16 at 17:03
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    $\begingroup$ @safesphere --Might a causal separation (or, at least, one side of it) qualify as an "initial condition"? (I've googled a paper or two whose publicly-accessible parts imply that it might.) $\endgroup$ – Edouard May 17 at 20:07
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    $\begingroup$ @Edouard Causal separation of what? Of matter. So your initial conditions include matter. $\endgroup$ – safesphere May 18 at 5:27
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No, general relativity doesn't make any claim as to whether matter must exist or not. In fact, the simplest of the solutions to the Einstein equations are vacuum solutions. For example, the Kerr-Newman blackholes and their special cases such as the Schwarzschild blackholes and Kerr blackholes. The dimensionality of spacetime is still $4$ in these solutions with one dimension being timelike. While these are all stationary solutions, you can also get non-stationary solutions in the vacuum. For example, gravitational waves. Gravitational waves are purely vacuum solutions and also exhibit non-trivial dynamics unlike the stationary solutions. So, the existence of time isn't contingent upon the existence of matter in general relativity.

I should clarify that in the case of Kerr-Newman blackholes, there exists electromagnetic fields so they are not truly vacuum solutions but still, they are solutions without the existence of any matter. Also, the special cases of the Kerr-Newman blackholes which are uncharged (i.e. the Schwarzschild blackholes and the Kerr blackholes) are truly vacuum solutions.

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    $\begingroup$ Physics is described by differential equations applied to realistic initial conditions. If you take unrealistic initial conditions and apply perfectly valid equations, you would get an unphysical result. Vacuum solutions are unphysical, they don’t exist in reality. Gravitational waves are created by matter just like light is created by electric charges, even if it then flies freely in space for billions of years. Your answer is incorrect. $\endgroup$ – safesphere May 16 at 6:54
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    $\begingroup$ @safesphere OK, if my answer is incorrect, can you point out exactly which postulate or theorem in general relativity requires that time doesn't exist without matter? $\endgroup$ – Dvij D.C. May 16 at 12:03
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    $\begingroup$ I didn’t mention GR. It is a general principle of physics that differential equations don’t describe the system, but the evolution of the system. You don’t just take the equations and derive physics from them. This is meaningless. Instead you take a real system and apply the equations to it to study its evolution. So vacuum solutions are meaningless, except for exterior Schwarzschild. The fact that time and energy are a canonical pair follows from the stationary action principle valid in classical and quantum physics, so the GR reference you are looking for is the stationary Hilbert action. $\endgroup$ – safesphere May 16 at 16:53
  • $\begingroup$ @Physics No, because comments are supposed to be primarily for discussing possible changes to the post and/or sometimes to provide some relevant supplementary information. safesphere's concerns are too broard, and at points, our disagreement might be considered a matter of opinion. For this reason, I don't think it's useful to continue the discussion (which is not what comments are for anyway). I'm always happy to discuss anything if anyone pings me in the chat. :) $\endgroup$ – Dvij D.C. May 17 at 20:42
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safesphere responded to my question by saying that "time and energy are Fourier conjugates (or more generally, spacetime and energy-momentum) and cannot exist in the physical reality without each other. In other words, GR states that spacetime is the field produced by matter just like the electromagnetic field is produced by charges."

My response is the following:

Spacetime and energy-momentum are not Fourier conjugates. In Newton mechanics they are topological duals (flows and generators).

This connection is different in curved spacetime. In curved spacetime the Fourier transform is not definable. This is the reason in mathematics some differential equations are perfectly understood in a flat space are not nearly understood even in simple curved spaces.

In GR it is not even possible to formulate the claim that spacetime and energy-momentum are Fourier conjugates.

There are many more things one could say here. There is a connection between spacetime and energy-momentum (this is the reason why energy is not conserved in GR). But only quantum observables in flat spacetime are connected via some kind Fourier transform. But even this connection is wrong for photons because they have no sharp spacetime observable but an energy-momentum observable.

As I said, I could say many more things here. But bottom line is, that in GR energy-momentum and spacetime are not Fourier conjugates.

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